10 research outputs found
Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids
This paper shows that every Plactic algebra of finite rank admits a finite
Gr\"obner--Shirshov basis. The result is proved by using the combinatorial
properties of Young tableaux to construct a finite complete rewriting system
for the corresponding Plactic monoid, which also yields the corollaries that
Plactic monoids of finite rank have finite derivation type and satisfy the
homological finiteness properties left and right . Also, answering a
question of Zelmanov, we apply this rewriting system and other techniques to
show that Plactic monoids of finite rank are biautomatic.Comment: 16 pages; 3 figures. Minor revision: typos fixed; figures redrawn;
references update
Garside and quadratic normalisation: a survey
Starting from the seminal example of the greedy normal norm in braid monoids,
we analyse the mechanism of the normal form in a Garside monoid and explain how
it extends to the more general framework of Garside families. Extending the
viewpoint even more, we then consider general quadratic normalisation
procedures and characterise Garside normalisation among them.Comment: 30 page
Logspace computations for Garside groups of spindle type
M. Picantin introduced the notion of Garside groups of spindle type,
generalizing the 3-strand braid group. We show that, for linear Garside groups
of spindle type, a normal form and a solution to the conjugacy problem are
logspace computable. For linear Garside groups of spindle type with homogenous
presentation we compute a geodesic normal form in logspace.Comment: 22 pages; short version as v1. Terminolgy and title changed. In
particular, in previous versions we called Garside groups of spindle type
"rigid Garside groups
Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory
We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research
A Homotopical Completion Procedure with Applications to Coherence of Monoids
International audienceOne of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the low-dimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting systems that keeps track of the cells generated by confluence diagrams. We extend the Knuth-Bendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids
Liftings of Nichols algebras of diagonal type III. Cartan type
We complete the classification of Hopf algebras whose infinitesimal braiding
is a principal Yetter-Drinfeld realization of a braided vector space of Cartan
type over a cosemisimple Hopf algebra. We develop a general formula for a
class of liftings in which the quantum Serre relations hold. We give a detailed
explanation of the procedure for finding the relations, based on the recent
work of Andruskiewitsch, Angiono and Rossi Bertone.Comment: 54 pages; including an appendix. Final version, to appear in J.
Algebr